Revenue, Cost, Price prob?

Question

In a yoyo shop, they found that they sold 1500 yoyos per week with a price of $2.50, but 2500 yoyos a week with a price of $2.00. Their fixed costs per week for the startup loan are $2000 and their cost to make each yoyo after paying that is $1.00
a) Find the demand function (price function)
b)Find the cost equation
c) Find the revenue equation
d) Find the profit equation
e) Find the price they should charge to maximize their weekly profit

Answer 1

a) From the given data, you are to obtain a linear relationship between the the price and the quantity sold at that price. Let p be the unit price, and let q denote the quantity of yoyos sold in a week at that price. Again, we assume a linear relationship:

q = Ap + B

where A and B are constants to be found. We're told that q=1500 when p=2.5, and q=2500 when p=2. So

1500 = A*2.5 + B
2500 = A*2 + B

You should have no trouble solving this system; A=-2000 and B=6500.

So

q = -2000p + 6500

Cost is fixed costs + (unit cost)*(number of units)

C(p) = 2000 + 1.0q = 2000 + (-2000p + 6500) = 8500 - 2000p

Revenue is (unit price)*(number of units sold)
(we assume they sell all that they make)

R(p) = pq = p(-2000p + 6500) = -2000p² + 6500p

Profit = Revenue - Cost

I'll leave the rest to you.

I get p=2.125 as the value of p that maximizes profit. The price has to be a whole number of cents, so either $2.12 or $2.13 (the two prices produce the same profit)